Optimal quantization methods for nonlinear filtering with discrete-time observations

TitleOptimal quantization methods for nonlinear filtering with discrete-time observations
Publication TypeJournal Article
Year of Publication2005
AuthorsGilles Pagès, and Huyên Pham
JournalBernoulli
Volume11(5)
KeywordsEuler scheme, Markov chain, nonlinear filtering, numerical approximation, stationary signal, stochastic gradient descent, vector quantization
Abstract

We develop an optimal quantization approach for numerically solving nonlinear filtering problems associated with discrete-time or continuous-time state process and discrete-time observations. Two quantization methods are proposed: a marginal quantization and a Markovian quantization of the signal process. The approximate filters are explicitly solved by a finite-dimensional forward procedure. A posteriori error bounds are stated and we show that the approximate error terms are minimal at some specific grids that may be computed off-line by a stochastic gradient method based on Monte Carlo simulations. Some numerical experiments are carried out: the convergence of the approximate filter as the accuracy of the quantization increases and its stability when the latent process is mixing are emphasized.